Q.A cubical block of density $\rho$ is floating on the surface of water. Out of its height $L$, fraction $x$ is submerged in water. The vessel is in an elevator accelerating upward with acceleration $a$. What is the fraction immersed?

Concept: Buoyancy and apparent gravity in an accelerating frame.

When the elevator accelerates upward with acceleration $a$, every object inside experiences an effective gravitational acceleration $g_{\text{eff}} = g + a$ instead of $g$. The buoyancy condition for a floating body remains unchanged in form: the weight of the displaced fluid equals the weight of the block, both calculated using the effective gravity.

For a cube of side $L$ and density $\rho$, floating with fraction $h/L$ submerged in water (density $\rho_w$):

Equating these:

$\rho L^3 g_{\text{eff}} = \rho_w L^2 h \cdot g_{\text{eff}}$ …

When a block floats, it displaces exactly enough fluid to balance its weight. The key insight is that both the weight of the block and the buoyant force scale with the same effective gravity in an accelerating elevator, so their ratio—and hence the equilibrium position—stays unchanged.

Why the fraction doesn't change

In the elevator's reference frame, every mass experiences an effective gravitational acceleration $g_{\text{eff}} = g + a$ (upward acceleration makes everything "heavier"). The block's effective weight becomes $\rho L^3 g_{\text{eff}}$, where $L^3$ is the volume of the cube. The buoyant force, which equals the weight of displaced water, becomes $\rho_{\text{water}} \cdot (\text{submerged volume}) \cdot g_{\text{eff}}$.

Notice that $g_{\text{eff}}$ appears in both expressions. When we write the floating condition, it cancels out completely.

Step-by-step derivation

1. Floating condition on Earth's surface (no acceleration) The block floats with fraction $x$ submerged. The submerged volume is $x L \cdot L^2 = x L^3$. Force balance:

Simplifying:

This tells us the density ratio: $x = \frac{\rho}{\rho_{\text{water}}}$.

2. In the accelerating elevator Replace $g$ with $g_{\text{eff}} = g + a$ everywhere. Let the new immersed fraction be $x'$, so the submerged volume is $x' L^3$. Force balance:

3. Cancel the common factor Divide both sides by $L^3(g + a)$:

This is identical to the equation from step 1. …

Accelerating lift: g_eff=g+a applies to both weight and buoyancy equally. rhoL^3g_eff=rho_wL^2h*g_eff -- g_eff cancels -- h/L=rho/rho_w, sam …